Pythagorean Theorem

Well, for starters, you ought to specify that we're referring to the equations . Also, this theorem is false: and must not only have opposite parities, they must be relatively prime. Example: and have opposite parity, yet is NOT primitive. The correct theorem should read: "If , then is primitive." Moving on...

To prove by contradiction, we start with the premise " and have no common divisors AND there exists a prime that divides ." We will show this premise to be impossible:

Lemma: if , and must have opposite parities, one even one odd. so , so or or . If then since and have opposite parities, so or . WLOG, suppose . Now, so . Since , so . Thus, , and we have our contradiction. Therefore our premise is false.